Answer
$1,1$ and $5,\dfrac{1}{5}$
Work Step by Step
Let $x$ be the number. Then its reciprocal is $\dfrac{1}{x}.$
The conditions of the problem translate to the equation
\begin{array}{l}\require{cancel}
x+5\cdot\dfrac{1}{x}=6
.\end{array}
Using the properties of equality, the equation above is equivalent to
\begin{array}{l}\require{cancel}
x+\dfrac{5}{x}=6
\\\\
x\left( x+\dfrac{5}{x} \right) =(6)x
\\\\
x^2+5=6x
\\\\
x^2-6x+5=0
\\\\
(x-5)(x-1)=0
.\end{array}
Equating each factor to zero and then solving for the variable, the solutions of the equation above are $
x=\left\{ 1,5 \right\}
.$
Hence the numbers are the pairs $
1,1$ and $5,\dfrac{1}{5}.$
